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arXiv:2408.12912 · Physical review. B./Physical review. B

Real-time control of non-Abelian anyons in Kitaev spin liquid under energy dissipation

Chihiro Harada, Atsushi Ono, Joji Nasu

Published Dec 18, 2024• 2 citations
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Abstract

Quantum spin liquids realized in the Kitaev model offer a platform for fractionalization of spin into two quasiparticles: itinerant Majoranas and localized visons. Introducing a uniform weak magnetic field associates a Majorana zero mode with each vison excitation. The vison accompanied by a Majorana zero mode is known to behave as a non-Abelian anyon, which has garnered significant attention for its potential applications in topological quantum computing. Although spatial and temporal control of these anyons is essential for exploring their applicability in quantum computing, numerical simulations of creating, moving, and annihilating non-Abelian anyons by an external field remain challenging as this field violates the exact solvability of the Kitaev model. Moreover, such a field to control non-Abelian anyons may disturb the quantum state due to the energy injection it causes. In this study, by introducing energy dissipation phenomenologically in real-time simulations, we demonstrate that the generation, movement, and annihilation of vison excitations can be achieved while maintaining their localization. In particular, we find that a vison can be moved in a desired direction by using time-dependent local magnetic fields or gradient magnetic fields, and it remains accompanied by a Majorana zero mode even after its movement. We also reveal that a larger spatial extent of the Majorana zero modes bound to a vison facilitates the movement of the vison with smaller field gradients. Furthermore, our numerical simulations demonstrate pair creation and annihilation of visons, triggered by time-dependent magnetic fields. The results obtained in this study highlight the significance of energy dissipation in controlling non-Abelian anyons, which will stimulate further investigations into the nonequilibrium dynamics of fractional quasiparticles in strongly correlated electron systems, as well as studies for applications in topological quantum computation.

DOI
10.1103/physrevb.110.214426

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Quantum spin liquids realized in the Kitaev model offer a platform for fractionalization of spin into two quasiparticles: itinerant Majoranas and localized visons. Introducing a uniform weak magnetic field associates a Majorana zero mode with each vison excitation. The vison accompanied by a Majorana zero mode is known to behave as a non-Abelian anyon, which has garnered significant attention for its potential applications in topological quantum computing. Although spatial and temporal control of these anyons is essential for exploring their applicability in quantum computing, numerical simulations of creating, moving, and annihilating non-Abelian anyons by an external field remain challenging as this field violates the exact solvability of the Kitaev model. Moreover, such a field to control non-Abelian anyons may disturb the quantum state due to the energy injection it causes. In this study, by introducing energy dissipation phenomenologically in real-time simulations, we demonstrate that the generation, movement, and annihilation of vison excitations can be achieved while maintaining their localization. In particular, we find that a vison can be moved in a desired direction by using time-dependent local magnetic fields or gradient magnetic fields, and it remains accompanied by a Majorana zero mode even after its movement. We also reveal that a larger spatial extent of the Majorana zero modes bound to a vison facilitates the movement of the vison with smaller field gradients. Furthermore, our numerical simulations demonstrate pair creation and annihilation of visons, triggered by time-dependent magnetic fields. The results obtained in this study highlight the significance of energy dissipation in controlling non-Abelian anyons, which will stimulate further investigations into the nonequilibrium dynamics of fractional quasiparticles in strongly correlated electron systems, as well as studies for applications in topological quantum computation.
The Kitaev quantum spin liquid possesses two fractional quasiparticles, itinerant Majorana fermions and localized visons. It provides a promising platform for realizing a Majorana zero mode trapped by a vison excitation. This local mode behaves as a non-Abelian anyon capable of applications to quantum computation. However, creating, observing, and manipulating visons remain challenging even in the pristine Kitaev model. Here, we propose a theory to control visons enabled by a time-dependent local magnetic field in the Kitaev spin liquid. Examining the time evolution of the magnetic state, we demonstrate that a vison follows a locally applied field sweeping in the system. We clarify that one can move a vison accompanied by a Majorana zero mode by choosing the velocity and shape of the local field appropriately. In particular, the controllability of visons using local fields shows nonlinear behavior for its strength, which originates from interactions between Majorana fermions and visons. The present results suggest that itinerant Majorana fermions other than zero modes play a crucial role in vison transport. Our finding will offer a guideline for controlling Majorana zero modes in the Kitaev quantum spin liquid.
Non-Abelian anyonic excitations of quantum spin liquids have potential for application to topological quantum computation, but designing logical operations requires developing protocols to faithfully create, move, and read out such quasiparticles. In this paper, we test two such protocols for manipulating Z 2 fluxes (“visons”) of ferromagnetic and antiferromagnetic Kitaev models perturbed by a small uniform magnetic field. We show that a protocol that employs a driving magnetic field locally applied to the sites of the lattice suffers from significant unavoidable errors. These errors, accumulating over time, prevent reaching high probabilities of generating and displacing flux pairs. However, individual spin couplings can also be turned on and off, thus it is possible to create and displace fluxes with nearly 100 % probability. We show that braiding non-Abelian anyons can be achieved with good precision with our second protocol. We comment on the feasibility of our protocols in solid-state devices and quantum emulators.
Understanding the nonequilibrium relaxation dynamics of quantum spin liquids is an important and relatively unexplored frontier, where fractionalized quasiparticles and emergent gauge fields can give rise to dynamical behaviors far beyond those of conventional magnets. Here we study the relaxation dynamics of the Kitaev honeycomb model following a thermal quench, focusing on visons—gapped Z2{{\mathbb{Z}}}_{2}Z2​ flux excitations of the emergent gauge field in the Kitaev spin liquid. The long-time evolution is governed by the diffusion and pair annihilation of visons, with both the activation barrier for vison motion and the effective vison-vison interactions mediated by fractionalized Majorana fermions. The resulting open-system quantum dynamics, induced by coupling to a thermal bath, can be efficiently simulated using an effective kinetic Monte Carlo approach. We uncover a rich set of temperature-dependent dynamical regimes. With decreasing temperature, the relaxation crosses over from diffusion-limited annihilation to terminal-velocity–limited kinetics, and ultimately to pronounced dynamical arrest and freezing. We show that this freezing behavior is closely tied to the formation of long-lived, metastable 3×3\sqrt{3}\times \sqrt{3}3​×3​ vison crystals, which strongly suppress further annihilation. Remarkably, even in the absence of true long-range crystalline order, the system exhibits a hidden coarsening dynamics of vison superclusters associated with an emergent Z3{{\mathbb{Z}}}_{3}Z3​ symmetry.
Non-Abelian anyons in quantum spin liquids (QSLs) provide a promising route to fault-tolerant topological quantum computation. In the exactly solvable Kitaev honeycomb model, such anyons of the QSL state can be bound to nonmagnetic spin vacancies and endowed with non-Abelian statistics by an infinitesimal magnetic field. Here, we investigate how this approach for stabilizing non-Abelian anyons extends to a finite magnetic field represented by a proper Zeeman term. Through large-scale density-matrix renormalization group simulations, we compute the vacancy-anyon binding energy as a function of magnetic field for both the ferromagnetic and antiferromagnetic Kitaev models. We find that anyon binding remains robust within the entire QSL phase for the ferromagnetic Kitaev model but breaks down already inside this phase for the antiferromagnetic Kitaev model. To compute a binding energy several orders of magnitude below the magnetic energy scale, we introduce both a refined definition and an extrapolation scheme based on carefully tailored perturbations.
The Kitaev model realizes a quantum spin liquid where the spin excitations are fractionalized into itinerant Majorana fermions and localized Z2{\mathbb{Z}}_{2}Z2​ vortices. Quantum entanglement between the fractional excitations can be utilized for decoherence-free topological quantum computation. Of particular interest is the anyonic statistics realized by braiding the vortex excitations under a magnetic field. Despite the promising potential, the practical methodology for creation and control of the vortex excitations remains elusive thus far. Here we theoretically propose how one can create and move the vortices in the Kitaev spin liquid. We find that the vortices are induced by a local modulation of the exchange interaction; especially, the local Dzyaloshinskii-Moriya (symmetric off-diagonal) interaction can create vortices most efficiently in the (anti)ferromagnetic Kitaev model, as it effectively flips the sign of the Kitaev interaction. We test this idea by performing the ab initio calculation for the candidate material $\ensuremath{\alpha}\text{\ensuremath{-}}\mathrm{Ru}{\mathrm{Cl}}_{3}$ through the manipulation of the ligand positions that breaks the inversion symmetry and induces the local Dzyaloshinskii-Moriya interaction. We also demonstrate a braiding of vortices by adiabatically and successively changing the local bond modulations.
The excitation spectrum of Kitaev quantum spin liquids consists of itinerant Majorana fermions, which can be gapless or gapped, and vortices of a Z2\mathbb{Z}_2Z2​ gauge field, known as visons, which are \new{gapped within a stable Z2\mathbb Z_2Z2​ spin liquid phase}. In this work, we investigate visons in Kitaev-type models where the Majorana fermions form a Fermi surface. In this case, the creation of a vison pair is analogous to introducing a local impurity potential in a metal. Since the gapless modes lead to strong finite-size effects, we compare the numerical calculation of the \new{two-vison gap} on finite lattices with the result from an analytical approach based on Green's function techniques. We find that the \new{two-vison gap} decreases as the size of the Fermi surface increases, signalling an instability of the quantum spin liquid ground state. We also show that larger Fermi surfaces tend to suppress the change in local spin correlations due to the Majorana-vison scattering potential.